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Or equivalently, why is jacobi identity equivalent to integrability of system? How do I understand it intuitively? Thanks.

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    $\begingroup$ Could you say more explicitly what you mean by the Jacobi identity and the holonomy of system? $\endgroup$ – Deane Mar 20 at 3:37
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    $\begingroup$ Once we make sense of the intended question, it's answered by Chapter II.3 of Baez-Muniain's book "Gauge fields, knots, and gravity". $\endgroup$ – Chris Gerig Mar 20 at 3:41